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    Re: Mars Tables
    From: Paul Hirose
    Date: 2008 Jun 03, 16:38 -0700

    This has been sitting in my Drafts folder for a week or so, and I
    finally got around to giving it the reply it deserves.
    
    Geoffrey Kolbe wrote:
    > Yes, I have. The accuracy of the fix depends on a number of factors. I have 
    > a good second order theodolite which will directly in seconds of arc. The 
    > scale cursor is suspended by a pendulum, which keeps the theodolite 
    > "upright" to a third of second of arc. But this instrument is effectively 
    > limited by its 30 power scope, which improves the 1 minute of arc naked eye 
    > resolving power, to about 2 seconds of arc - in good light with good 
    > contrast. At night, with the star moving rapidly across the field of view 
    > and the cross hairs dimly illuminated, it is a struggle to get single sight 
    > altitudes much better than about 3 seconds of arc in my experience. Doing 
    > better seems to rely on the use of 'impersonal' observation methods, for 
    > example a timed camera on the eyepiece and a graduated reticule.
    
    What theodolite do you own?
    
    I'm not sure more power would increase accuracy, since the apparent
    motion of the star already gives you significant difficulty at 30x.
    
    The highest power eyepiece on a Wild T3 (a first order theodolite
    reading to tenth seconds) gives 40x. That's not much more than you have.
    Remember that accurate pointing with a theodolite is a matter of
    centering a symmetrical target in the crosshairs, not resolving fine
    detail in the target.
    
    There is a way to test for a proper match between telescope
    magnification and the angle reading resolution: 1) Center the crosshair
    on a well defined target. 2) Bring the circle reading micrometer into
    coincidence. 3) Move the crosshair off target so the pointing error is
    just noticeable. 4) If the instrument is well designed, the micrometer
    should likewise have a just noticeable coincidence error.
    
    My Russian instrument is in the same class as yours, and it passes this
    test.
    
    When an index level for the vertical circle is present (the compensators
    in our theodolites make this unnecessary), the same principle applies.
    A just noticeable movement of the bubble should cause a just noticeable
    change in the vertical angle reading.
    
    A lack of proper balance forces you to exercise undue care with one part
    of the instrument in order to realize the accuracy potential of some
    other part.
    
    > Of course, even with this level of accuracy, one starts to run into 
    > problems as to which geoid you are using for your coordinate system. For 
    > example. If you stand on the centre line of the transit telescope in 
    > Flamstead House at Greenwich with a GPS receiver, you will find that it 
    > gives a WGS84 longitude of a little over 5 seconds of arc to the West. So, 
    > having found a fix good to 1 second of arc according to the geoid assumed 
    > for celestial navigation, you would need to transform this to the 
    > terrestrial reference frame being used by the map on which you were trying 
    > to find my position.
    
    I think you mean "datum", which is a set of constants used to calculate
    the latitude and longitude of a place. On the other hand, the geoid is
    the theoretical surface that is the basis for height above sea level.
    It's also the gravitational level surface that sextants and theodolites
    use. You can't pick your geoid. At any given place, it is what it is.
    
    Due to irregularities in Earth's mass distribution, the geoid is an
    irregular surface. It differs slightly from the mathematically perfect
    ellipsoid that is the basis for a datum. So, a plumb line does not stand
    exactly perpendicular to the ellipsoid. This discrepancy is called
    "deflection of the vertical". It causes gravity referenced instruments
    (sextants, theodolites) to generate "astronomic" latitudes and
    longitudes, which are slightly different from the "geodetic" coordinates
    shown on maps and GPS receivers.
    
    In extreme cases deflection of the vertical can reach one minute of arc.
    Values of several seconds are common. In Los Angeles, the NS component
    is 18" and the EW component 5". This is definitely significant to a 
    1-second theodolite.
    
    Conversion from astronomic to geodetic coordinates is straightforward if
    the deflection of the vertical components are known. The formulas are:
    
    φ = Φ - ξ
    λ = Λ - η / cos Φ
    
    where
    
    φ, λ are the geodetic latitude and longitude, respectively
    Φ, Λ are the astronomic latitude and longitude
    ξ, η are the north and east deflections of the vertical (they are
    positive if the plumb line intersects the celestial sphere north or east
    of the ellipsoidal normal)
    
    The north component is sometimes called ζ (zeta).
    
    With a minor rearrangement those formulas will give astronomic
    coordinates, useful to predict the azimuth and elevation of stars in the
    "deflected" horizontal frame seen by a theodolite, for example. The
    procedure is to use the observer's astronomic latitude and longitude for
    the step that converts the star coordinates from the topocentric
    equatorial frame to the horizontal frame.
    
    Use the observer's geodetic coordinates for all other parts of the
    calculation (hour angle, etc.), because these depend entirely on where
    the observer is, not the direction of the gravity vector at that place.
    
    For the practical sextant navigator these refinements are a waste of
    time. (Where would you get deflection of the vertical values for
    locations at sea?) But perhaps an experimenter, working from the same
    place and keen on accounting for all systematic effects, might take
    pains to convert the site's GPS position to astronomic coordinates.
    
    -- 
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